manaman said:
.999... = 1 however isn't a trick. But makes sense when you really start to think about it.
It is indeed true. If you express the value, instead as the sum of 9(.1)^n, where n ranges from 1 to infinity, you have an infinite arithmetic series. Since math at infinity is not directly possible, we can examine what happens as n approaches infinity. In this case, the limit simply approaches 1, meaning that, after the sum of this infinite series is, indeed 1.
A better way to describe it that does not involve the dark magicks of calculus uses simple algebra. Remember for a moment the fundamental theorem of Algebra: An equation of nth degree has exactly n solutions.
The series 0.999... will stand in for n.
10n = 9.999... (the trailing elapses indicates the series continues to infinity)
10n - n = 9.999... - 0.999...
9n = 9
n = 1
The fundamental theorem of algebra thus indicates that .999... is equivalent to 1 as this first degree equation must have one solution.
It should be mentioned as a caution that mathematical fallacy is commonly used in an attempt to prove silly things like 2 = 1. In many cases, the fallacy lies in a non-obvious division by zero somewhere hidden in the middle of the associated proof.
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